Optimal. Leaf size=64 \[ -\frac{3 x}{-x^2+e^{2 i a}}+\frac{e^{2 i a}}{x \left (-x^2+e^{2 i a}\right )}-2 e^{-i a} \tanh ^{-1}\left (e^{-i a} x\right ) \]
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Rubi [F] time = 0.0478845, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\cot ^2(a+i \log (x))}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\cot ^2(a+i \log (x))}{x^2} \, dx &=\int \frac{\cot ^2(a+i \log (x))}{x^2} \, dx\\ \end{align*}
Mathematica [A] time = 0.121487, size = 72, normalized size = 1.12 \[ \frac{2 x (\cos (a)-i \sin (a))}{\left (x^2-1\right ) \cos (a)-i \left (x^2+1\right ) \sin (a)}-2 \cos (a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))+2 i \sin (a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))+\frac{1}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 62, normalized size = 1. \begin{align*}{x}^{-1}-2\,{\frac{1}{x \left ( \left ({{\rm e}^{i \left ( a+i\ln \left ( x \right ) \right ) }} \right ) ^{2}-1 \right ) }}-{\frac{\ln \left ({{\rm e}^{ia}}+x \right ) }{{{\rm e}^{ia}}}}+{\frac{\ln \left ({{\rm e}^{ia}}-x \right ) }{{{\rm e}^{ia}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15388, size = 385, normalized size = 6.02 \begin{align*} -\frac{2 \,{\left ({\left (i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) +{\left (i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right )\right )} x^{3} +{\left ({\left (2 \,{\left (-i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \cos \left (2 \, a\right ) +{\left (2 \, \cos \left (a\right ) - 2 i \, \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) +{\left (2 \,{\left (-i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \cos \left (2 \, a\right ) +{\left (2 \, \cos \left (a\right ) - 2 i \, \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right )\right )} x - 6 \, x^{2} +{\left (x^{3}{\left (\cos \left (a\right ) - i \, \sin \left (a\right )\right )} -{\left ({\left (\cos \left (a\right ) - i \, \sin \left (a\right )\right )} \cos \left (2 \, a\right ) +{\left (i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} x\right )} \log \left (x^{2} + 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) -{\left (x^{3}{\left (\cos \left (a\right ) - i \, \sin \left (a\right )\right )} -{\left ({\left (\cos \left (a\right ) - i \, \sin \left (a\right )\right )} \cos \left (2 \, a\right ) -{\left (-i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} x\right )} \log \left (x^{2} - 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) + 2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )}{2 \, x^{3} - x{\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (x e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x\right )}{\rm integral}\left (-\frac{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1}{x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x^{2}}, x\right ) - 2}{x e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.00885, size = 44, normalized size = 0.69 \begin{align*} \frac{3 x^{2} - e^{2 i a}}{x^{3} - x e^{2 i a}} - \left (- \log{\left (x - e^{i a} \right )} + \log{\left (x + e^{i a} \right )}\right ) e^{- i a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25604, size = 117, normalized size = 1.83 \begin{align*} 2 \,{\left (\frac{\arctan \left (\frac{x}{\sqrt{-e^{\left (2 i \, a\right )}}}\right ) e^{\left (-2 i \, a\right )}}{\sqrt{-e^{\left (2 i \, a\right )}}} + \frac{x e^{\left (-2 i \, a\right )}}{x^{2} - e^{\left (2 i \, a\right )}}\right )} e^{\left (2 i \, a\right )} + \frac{5 \, x^{2}}{x^{3} - x e^{\left (2 i \, a\right )}} - \frac{e^{\left (2 i \, a\right )}}{x^{3} - x e^{\left (2 i \, a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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